# Honors Algebra 2: Extra Practice on Unit 7

• Class: Honors Algebra 2
• Author: Peter Atlas
• Algebra and Trigonometry: Structure and Method, Brown

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Simplify Completely:

1. $$\displaystyle i^{1937} + i^{1938} + i^{1939} =$$
2. Solution

$$\displaystyle -1$$

3. $$\displaystyle 4\sqrt{-\frac{7}{12}} - 3 \sqrt{-\frac{1}{24}}$$
4. Solution

$$\displaystyle -\frac{i\sqrt{6}}{4} + \frac{2i\sqrt{21}}{3}$$

5. $$\displaystyle \left( 3 - i \right) ^{-2}$$
6. Solution

$$\displaystyle \frac{2}{25} + \frac{3}{50}i$$

7. Given $$\displaystyle a = -\frac{5}{6} + \frac{1}{2}i$$ and $$\displaystyle b = \frac{2}{3} + \frac{3}{4}i,$$ find $$\overline{a} + b$$
8. Solution

$$\displaystyle -\frac{1}{6} + \frac{1}{4}i$$

9. Factor $$\displaystyle x^4 + 5x^2 + 6$$ over the complex numbers
10. Solution

$$\displaystyle \left( x + i\sqrt{3} \right) \left( x - i\sqrt{3} \right) \left( x + i\sqrt{2}\right) \left( x - i\sqrt{2}\right)$$

11. Given $$\displaystyle z = -4 + 3i,$$ express $$\displaystyle z^{-1}$$ as a complex number
12. Solution

$$\displaystyle -\frac{4}{25} - \frac{3}{25}i$$

13. Solve for $$x$$ and $$y:$$ $$\displaystyle \frac{1}{2}x - 5y - \left(7y - \frac{3}{2}x\right)i = 32 + 45i$$
14. Solution

$$\displaystyle x = \frac{1}{4}, y=-\frac{51}{8}$$

15. Solve for real values of $$x$$ and $$y:$$ $$\displaystyle 3y + yi = 2i - xi + x$$
16. Solution

$$\displaystyle x = \frac{3}{2}; y = \frac{1}{2}$$

17. Find the sum and difference of $$\displaystyle 7 + 5i$$ and its conjugate.
18. Solution

Sum: 14; Difference: $$10i$$

19. Express each of the following as a complex number given $$\displaystyle a = 5 - 8i; b = -7 + 3i; c = 2 + 8i; d = -4i$$
1. $$a + b$$
2. Solution

$$\displaystyle -2 - 5i$$

3. $$a + c$$
4. Solution

7

5. $$a - c$$
6. Solution

$$\displaystyle 3 - 16i$$

7. $$c - d$$
8. Solution

$$\displaystyle 2 + 12i$$

9. $$\overline{c} - d$$
10. Solution

$$\displaystyle 2 - 4i$$

11. $$d - \overline{a}$$
12. Solution

$$\displaystyle -5 - 12i$$

20. Factor completely over the complex numbers: $$\displaystyle a^2 + 4$$
21. Solution

$$\displaystyle (a + 2i)(a - 2i)$$

22. Factor completely over the complex numbers: $$\displaystyle x^4 + 7x^2 + 12$$
23. Solution

$$\displaystyle \left(x + 2i\right)\left(x - 2i\right)\left(x + i\sqrt{3}\right)\left(x - i\sqrt{3}\right)$$

24. Express the reciprocal of $$\displaystyle 2 - 5i$$ as a complex number.
25. Solution

$$\displaystyle \frac{2}{29} + \frac{5}{29}i$$

26. Express $$\displaystyle \frac{2 + 3i}{7 + 4i}$$ as a complex number.
27. Solution

$$\displaystyle \frac{2}{5} + \frac{1}{5}i$$

28. $$\displaystyle (4 - 7i)(3 + i) =$$
29. Solution

$$\displaystyle 19 - 17i$$

30. $$\displaystyle \left(3 - 4i \right)^2 =$$
31. Solution

$$\displaystyle -7 - 24i$$

32. $$\displaystyle \left(-1 + i\sqrt{5}\right)^2 =$$
33. Solution

$$\displaystyle -4 - 2i\sqrt{5}$$

34. $$\displaystyle \frac{3 + i}{4 - i}$$
35. Solution

$$\displaystyle \frac{11}{17} + \frac{7}{17}i$$

36. $$\displaystyle \frac{8i}{1 + 3i}$$
37. Solution

$$\displaystyle \frac{12}{5} + \frac{4}{5}i$$

38. $$\displaystyle \frac{3 - 6i}{-2 - 5i}$$
39. Solution

$$\displaystyle \frac{24}{29} + \frac{27}{29} i$$